Question

What is the value of \(\int_0^∞ e^{-x^2}  dx\)?

(a) \( \sqrt{\pi} \)

(b) \(\frac{\sqrt{\pi}}{\sqrt{2}} \)

(c) \(\frac{\sqrt{\pi}}{2} \)

(d) \(\frac{\pi}{2} \)

The question was posed to me by my college director while I was bunking the class.

Enquiry is from Special Functions topic in division Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

The correct option is (c) \(\frac{\sqrt{\pi}}{2} \)

The best I can explain: Substitute \(x^2 = y\)

\(2xdx = dy\)

= \(\int_0^∞ x^{\frac{-1}{2}}  e^{-x} dx \)

This is of the form of Gamma function. Here, \(n-1 = \frac{-1}{2} \). Therefore \(n = \frac{1}{2}.\)

Therefore, \(\Gamma(\frac{\frac{1}{2}}{2}) = \frac{\sqrt\pi}{2}.\)